Electron Diffraction

Spec mapping: OCR H556 Module 4.5 — Quantum physics (experimental evidence for the wave nature of electrons; the use of crystal lattices as natural diffraction gratings; relationship between accelerating potential difference and diffraction-ring radius; applications of electron diffraction including the electron microscope). Refer to the official OCR H556 specification document for exact wording.

De Broglie's hypothesis of matter waves was — in 1924 — a striking theoretical conjecture with no direct experimental support. Within three years, that changed. In 1927, two independent experiments confirmed that electrons do indeed diffract like waves, with wavelengths in precise agreement with the de Broglie formula. The first was by Clinton Davisson and Lester Germer at Bell Laboratories in New Jersey; the second was by George Paget Thomson at the University of Aberdeen. Both Davisson and Thomson shared the Nobel Prize in 1937.

(A delicious historical detail: George Paget Thomson's father, J. J. Thomson, had received the 1906 Nobel Prize for discovering the electron — as a particle. Father proved the electron was a particle; son proved it was a wave.)

This lesson examines the experimental evidence for the wave nature of matter, the techniques used, and the applications of electron diffraction in modern science. It is a key part of Module 4.5 in the OCR A-Level Physics A specification (H556).

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Why Diffraction Is the Decisive Test

Diffraction and interference are the hallmarks of wave behaviour. When a wave passes through an aperture or an array of obstacles comparable in size to its wavelength, it spreads out and produces a characteristic pattern of maxima and minima. Classical particles do not do this: they travel in straight lines and do not produce interference patterns.

So if you can make electrons diffract, you have shown they have wave properties. And if you can measure the diffraction pattern and extract a wavelength from it, you can compare that wavelength with the de Broglie prediction λ = h/p.

The trouble is: the electrons produced in ordinary laboratories have de Broglie wavelengths of order 10⁻¹⁰ m — about the size of a single atom. You cannot machine slits of this size. The only "gratings" with spacings of this order are crystals, where the regular spacing of atoms in a crystal lattice acts as a naturally occurring diffraction grating.

Crystals had already been used as X-ray diffraction gratings since 1912 (the work of von Laue and the Braggs), so the technique was well understood. Davisson and Germer accidentally discovered electron diffraction while studying electron scattering from nickel. Thomson deliberately passed electron beams through thin polycrystalline metal foils.

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The Davisson–Germer Experiment (1927)

The experimental set-up:

flowchart LR
    G[Electron gun] --> B[Collimated beam]
    B --> C[Nickel crystal]
    C --> D[Scattered electrons]
    D --> DET[Movable detector]
    DET --> M[Current meter]

• Electron gun: a heated tungsten filament emits electrons by thermionic emission. These are accelerated through a variable potential difference V, giving each electron kinetic energy eV and therefore momentum p = √(2 m eV).
• Target: a single crystal of nickel, cut and polished so that one of the crystal planes faces the incoming beam.
• Detector: a movable Faraday cup, which measures the scattered electron current as a function of scattering angle θ.

What Davisson and Germer found was spectacular. At particular values of the accelerating voltage V and scattering angle θ, the detected current showed strong maxima. These maxima could not be explained by classical particle scattering; they had the precise angular spacing predicted by a wave diffracting from the crystal lattice.

Specifically, for a nickel crystal with lattice spacing d and a normal-incidence electron beam, constructive interference occurs when

$$d sin \theta = n \lambda$$dsinθ=nλ

(the same formula as for a diffraction grating — in fact, the two-dimensional surface of the crystal was acting as a two-dimensional grating). At V = 54 V, a strong maximum appeared at θ = 50°. From the known lattice spacing d of nickel, the corresponding wavelength was

$$\lambda = d sin \theta = (2.15 \times 10^{-10})(sin 50^{\circ}) \approx 1.65 \times 10^{-10} m$$λ=dsinθ=(2.15×10−10)(sin50∘)≈1.65×10−10m

And the de Broglie prediction from λ = h/√(2 m eV) with V = 54 V:

$$\lambda = 1.226 \times 10^{-9}/ \sqrt{54} \approx 1.67 \times 10^{-10} m$$λ=1.226×10−9/54​≈1.67×10−10m

Agreement to within experimental error. The electron wavelength was exactly what de Broglie had predicted.

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The Thomson Experiment (1927)

Working independently in Aberdeen, George Paget Thomson used a different geometry. He passed a beam of electrons through a thin polycrystalline metal foil (initially a thin film of celluloid, later metal foils of gold, aluminium, etc.) and caught the transmitted electrons on a photographic plate.

flowchart LR
    G[Electron gun] --> V[High voltage]
    V --> B[Beam through thin foil]
    B --> F[Polycrystalline metal foil]
    F --> R[Diffracted rings]
    R --> P[Photographic plate / fluorescent screen]

Because the foil was polycrystalline (composed of many small crystal grains oriented in random directions), the diffraction pattern was not a set of discrete spots but a series of concentric rings — one ring for each allowed diffraction order from each family of crystal planes. The radii of the rings could be measured and used to infer the electron wavelength.

Again, the wavelengths obtained agreed perfectly with de Broglie's formula λ = h/p.

Thomson's ring pattern was visually striking — it looked exactly like the powder-X-ray-diffraction patterns that crystallographers had been producing for years, but made with electrons. For anyone who was still unconvinced after Davisson–Germer, Thomson's photographs were the clincher.

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The Appearance of the Pattern

A modern electron-diffraction tube in an A-Level laboratory uses a thin film of graphite as the target. Graphite has a layered hexagonal structure; when a beam of electrons is passed through it, two sets of ring patterns appear, corresponding to two families of crystal planes. The radii of the rings are:

$$r = L tan(2 \theta )$$r=Ltan(2θ)

where L is the distance from the foil to the screen and 2θ is the scattering angle (the factor of 2 is because the beam is reflected off the crystal plane with angle of incidence θ equal to angle of reflection).

Key observations:

1. At higher accelerating voltages, the electrons have higher momentum, hence shorter de Broglie wavelength. The diffraction rings become smaller (they collapse towards the centre). This is exactly analogous to optical diffraction: for a fixed grating, shorter wavelengths give narrower-angle diffraction.
2. The ring pattern is sharp and reproducible, just like optical diffraction rings.
3. Individual electrons register as single points on the fluorescent screen, consistent with particle-like detection. But the statistical pattern of many electrons shows unmistakable wave-like interference.

flowchart LR
    L1["Low V<br/>large p<br/>NO — small p → large λ"] --> L2["Wrong"]
    H1["High V"] --> H2["High p"]
    H2 --> H3["Small λ"]
    H3 --> H4["Small rings"]

    L3["Low V"] --> L4["Low p"]
    L4 --> L5["Large λ"]
    L5 --> L6["Large rings"]

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Worked Example: Electron Diffraction from Graphite

In an A-Level electron-diffraction apparatus, a beam of electrons is accelerated through 4.0 kV and passes through a thin graphite film to a screen 13.5 cm away. The first diffraction ring has a radius of 2.8 cm.

(a) Calculate the de Broglie wavelength of the electrons.

(b) Using 2d sin θ = nλ with n = 1, find the atomic plane spacing d of the graphite, given that the scattering angle 2θ is determined by the geometry.

Solution.

(a) Accelerated through 4.0 kV, so kinetic energy = eV = 4.0 × 10³ eV. Using the shortcut:

$$\lambda = 1.226 \times 10^{-9}/ \sqrt{V} = 1.226 \times 10^{-9}/ \sqrt{4000} \approx 1.94 \times 10^{-11} m$$λ=1.226×10−9/V​=1.226×10−9/4000​≈1.94×10−11m

About 19 pm — much shorter than the visible-light or X-ray wavelengths we're used to.

(b) The scattering angle geometry: if the ring has radius r = 2.8 cm at distance L = 13.5 cm, then

$$\begin{aligned} tan(2 \theta ) = r/L = 2.8/13.5 = 0.207 \\ 2 \theta = 11.7^{\circ} \\ \theta = 5.85^{\circ} \end{aligned}$$tan(2θ)=r/L=2.8/13.5=0.2072θ=11.7∘θ=5.85∘​

From Bragg's law, 2d sin θ = λ:

$$\begin{aligned} d = \lambda /(2 sin \theta ) \\ = (1.94 \times 10^{-11})/(2 \times sin 5.85^{\circ}) \\ = (1.94 \times 10^{-11})/(2 \times 0.102) \\ = 9.5 \times 10^{-11} m \\ \approx 0.095 nm \end{aligned}$$d=λ/(2sinθ)=(1.94×10−11)/(2×sin5.85∘)=(1.94×10−11)/(2×0.102)=9.5×10−11m≈0.095nm​

This is the order of magnitude of the carbon-carbon layer spacing in graphite, which is about 0.34 nm — so our answer is within a factor of a few, consistent with a higher-order diffraction or a different set of planes. (The first, strongest ring usually corresponds to the in-plane spacing a ≈ 0.12 nm, which is closer to our answer.)

Answer: λ ≈ 19 pm; d ≈ 0.09–0.12 nm depending on which ring is observed.

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Electron Microscopes

Electron diffraction is not just a curiosity — it is the working principle of one of the most important instruments in modern science: the electron microscope.

An ordinary optical microscope is limited by the wavelength of visible light (~500 nm) to resolving details no smaller than about 200 nm. Two objects closer together than this cannot be distinguished as separate; they blur into one. This limit comes from diffraction: any optical instrument is fundamentally restricted by

$$d_{\text{min}} \approx \lambda /(2 NA)$$dmin​≈λ/(2NA)

where NA is the numerical aperture of the lens. There is no way to do better with visible light.

An electron microscope substitutes the electron beam for light. Electrons accelerated through, say, 100 kV have a de Broglie wavelength of

$$\lambda = 1.226 \times 10^{-9}/ \sqrt{10^{5}} \approx 3.9 \times 10^{-12} m$$λ=1.226×10−9/105​≈3.9×10−12m

— about 4 picometres, roughly 100 000 times shorter than visible light. The corresponding diffraction limit is roughly 100 000 times smaller — comfortably less than the size of an atom.

Modern electron microscopes come in two main varieties:

1. Transmission Electron Microscope (TEM): electrons pass through an ultra-thin sample, form an image via magnetic lenses, and are captured on a fluorescent screen or CCD. Resolution: better than 0.1 nm — good enough to image individual atoms.
2. Scanning Electron Microscope (SEM): a focused electron beam is scanned across the surface of a solid sample, and secondary or backscattered electrons are detected to build up an image pixel by pixel. Resolution: roughly 1 nm — coarser than TEM, but gives striking three-dimensional surface images.

Both types of microscope depend on the wave-particle duality of electrons. The electron diffracts like a wave in the lens system (the image-forming element is a set of magnetic lenses that bend the electron beam exactly as glass lenses bend light), but it is detected like a particle when it hits the screen or sensor.

Without de Broglie — and Davisson, Germer and Thomson's experimental confirmation — there would be no electron microscopes, no detailed images of viruses, proteins, single atoms, or nanostructures. Much of modern biology, chemistry and materials science depends directly on wave-particle duality.

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Electron Diffraction in Materials Science

Beyond microscopy, electron diffraction is used routinely to determine the structure of crystals. The technique is called selected-area electron diffraction (SAED): an electron beam is directed at a chosen region of a sample, and the diffraction pattern is recorded. The positions and intensities of the diffraction spots reveal the lattice geometry, crystal symmetry, and in many cases the orientation of individual grains.

Electron diffraction complements X-ray diffraction in several ways:

• Electrons interact much more strongly with matter than X-rays do, so much thinner samples (nanometres) can be studied.
• Electron beams can be focused to very small spots, so local structure can be probed with nanometre spatial resolution.
• Electrons can be used to study the surfaces of materials, which is difficult with X-rays.

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Electron Diffraction Pattern — Visual Reference

The following SVG shows the characteristic ring pattern on a fluorescent screen of an A-Level electron-diffraction tube using a graphite target. Two ring radii are shown, corresponding to two atomic-plane families in graphite. Higher accelerating voltage shrinks the rings towards the centre.

The bright central spot is the un-deflected beam; the two faint rings are the constructive-interference maxima from two families of atomic planes in the graphite. The ring radii are given by $r = L \tan(2\theta)$r=Ltan(2θ) where $\theta$θ satisfies Bragg's law $2 d \sin\theta = n\lambda$2dsinθ=nλ.

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Synoptic Links

• de Broglie wavelength (Module 4.5, previous lesson) — electron diffraction is the experimental test of $\lambda = h/\sqrt{2 m_e e V}$λ=h/2me​eV​. The agreement between measured ring radii and predicted wavelengths is the empirical foundation of matter waves.
• Bragg's law (post-A-Level X-ray crystallography) — $2d\sin\theta = n\lambda$2dsinθ=nλ applies to both X-rays and electron waves diffracting from crystal planes. A-Level treats this as a transferable rule from the diffraction-grating equation $d\sin\theta = n\lambda$dsinθ=nλ applied to crystal planes; the factor of 2 arises from the back-and-forth path through the layer.
• Capacitors and charged-particle acceleration (Module 6.1) — the electron gun is a parallel-plate accelerator: $W = eV$W=eV supplies KE to the electron, which then becomes the de Broglie momentum $p = \sqrt{2 m_e e V}$p=2me​eV​.
• Resolution limits of optical instruments (Module 4.4) — the diffraction limit $d_{\min} \approx \lambda/(2 NA)$dmin​≈λ/(2NA) applies to any wave-based instrument. Electron microscopes beat optical microscopes by ~$10^5$105 because their (de Broglie) wavelength is ~$10^5$105 smaller.

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Specimen question modelled on the OCR H556 paper format

Question (10 marks): In an A-Level electron-diffraction tube, electrons are accelerated through a potential difference $V$V and pass through a thin graphite film onto a fluorescent screen at distance $L = 0.140$L=0.140 m.

(a) State and explain one observation from such an experiment that provides evidence for the wave nature of electrons. [2]

(b) The accelerating voltage is set to $V = 4500$V=4500 V. (i) Calculate the de Broglie wavelength of the electrons. (ii) The first-order ring has radius $r = 1.8$r=1.8 cm. Calculate the corresponding atomic-plane spacing $d$d of the graphite. (Use the small-angle / Bragg relation $\lambda \approx 2 d \theta$λ≈2dθ with $\theta = r/(2L)$θ=r/(2L) as a reasonable approximation.) [5]

(c) State and explain how the appearance of the rings changes when the accelerating voltage is doubled. Comment on whether your reasoning supports the wave model or the particle model. [3]

AO breakdown